A metric tangential calculus

Elisabeth Burroni and Jacques Penon

The metric jets, introduced here, generalize the jets (at order one) of
Charles Ehresmann. In short, for a ``good'' map f (said to be
``tangentiable'' at a) between metric spaces, we define its metric jet
tangent at a (composed of all the maps which are locally lipschitzian at
a and tangent to f at a) called the ``tangential'' of f at a,
and denoted Tf_a. So, in this metric context, we define a ``new
differentiability'' (called ``tangentiability'') which extends the
classical differentiability (while preserving most of its properties) to
new maps, traditionally pathologic.